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G = D4.5D8order 128 = 27

5th non-split extension by D4 of D8 acting via D8/C8=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: D4.5D8, Q8.5D8, C16.24D4, M4(2).36D4, M5(2).28C22, D4○C162C2, C4.43(C2×D8), (C2×SD32)⋊2C2, C4○D4.28D4, C8.6(C4○D4), C8.107(C2×D4), D4.4D4.C2, C8.4Q87C2, M5(2)⋊C28C2, D4.5D43C2, C8.17D48C2, C8○D4.9C22, C4.98(C4⋊D4), C2.26(C87D4), (C2×C8).240C23, (C2×C16).28C22, (C2×D8).50C22, C22.8(C4○D8), C8.C4.7C22, (C2×Q16).49C22, (C2×C4).45(C2×D4), SmallGroup(128,955)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — D4.5D8
C1C2C4C2×C4C2×C8C8○D4D4○C16 — D4.5D8
C1C2C4C2×C8 — D4.5D8
C1C2C2×C4C8○D4 — D4.5D8
C1C2C2C2C2C4C4C2×C8 — D4.5D8

Generators and relations for D4.5D8
 G = < a,b,c,d | a4=b2=1, c8=d2=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c7 >

Subgroups: 180 in 72 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×3], C8 [×2], C8 [×3], C2×C4, C2×C4 [×2], D4, D4 [×3], Q8, Q8 [×2], C23, C16 [×2], C16, C2×C8, C2×C8, M4(2), M4(2) [×3], D8 [×3], SD16 [×2], Q16 [×3], C2×D4, C2×Q8, C4○D4, C4.D4, C4.10D4, C8.C4 [×2], C2×C16, C2×C16, M5(2), M5(2), SD32 [×2], C8○D4, C2×D8, C2×Q16, C8⋊C22, C8.C22, M5(2)⋊C2, C8.17D4, C8.4Q8, D4.4D4, D4.5D4, D4○C16, C2×SD32, D4.5D8
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D8 [×2], C2×D4 [×2], C4○D4, C4⋊D4, C2×D8, C4○D8, C87D4, D4.5D8

Character table of D4.5D8

 class 12A2B2C2D4A4B4C4D8A8B8C8D8E8F8G16A16B16C16D16E16F16G16H16I16J
 size 112416224162244416162222444444
ρ111111111111111111111111111    trivial
ρ2111-1-111-1-1111-1-11111111-1-1-1-11    linear of order 2
ρ3111-1-111-11111-1-1-11-1-1-1-1-11111-1    linear of order 2
ρ411111111-111111-11-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ51111-11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 2
ρ6111-1111-1-1111-1-11-1-1-1-1-1-11111-1    linear of order 2
ρ7111-1111-11111-1-1-1-111111-1-1-1-11    linear of order 2
ρ81111-1111-111111-1-11111111111    linear of order 2
ρ922-2002-20022-20000-2-2-2-2200002    orthogonal lifted from D4
ρ10222202220-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ1122-2002-20022-200002222-20000-2    orthogonal lifted from D4
ρ12222-2022-20-2-2-222000000000000    orthogonal lifted from D4
ρ1322-220-22-200000000-22-22-2-22-222    orthogonal lifted from D8
ρ1422-2-20-222000000002-22-22-22-22-2    orthogonal lifted from D8
ρ1522-220-22-2000000002-22-222-22-2-2    orthogonal lifted from D8
ρ1622-2-20-22200000000-22-22-22-22-22    orthogonal lifted from D8
ρ1722200-2-2000002i-2i002-22-2-2--2-2-2--22    complex lifted from C4○D8
ρ1822-2002-200-2-220000000002i2i-2i-2i0    complex lifted from C4○D4
ρ1922200-2-200000-2i2i002-22-2-2-2--2--2-22    complex lifted from C4○D8
ρ2022200-2-2000002i-2i00-22-222-2--2--2-2-2    complex lifted from C4○D8
ρ2122-2002-200-2-22000000000-2i-2i2i2i0    complex lifted from C4○D4
ρ2222200-2-200000-2i2i00-22-222--2-2-2--2-2    complex lifted from C4○D8
ρ234-4000000022-22000001613+2ζ16111615+2ζ169165+2ζ163167+2ζ16000000    complex faithful
ρ244-4000000022-2200000165+2ζ163167+2ζ161613+2ζ16111615+2ζ169000000    complex faithful
ρ254-40000000-2222000001615+2ζ169165+2ζ163167+2ζ161613+2ζ1611000000    complex faithful
ρ264-40000000-222200000167+2ζ161613+2ζ16111615+2ζ169165+2ζ163000000    complex faithful

Smallest permutation representation of D4.5D8
On 32 points
Generators in S32
(1 13 9 5)(2 14 10 6)(3 15 11 7)(4 16 12 8)(17 21 25 29)(18 22 26 30)(19 23 27 31)(20 24 28 32)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32)
(1 8 9 16)(2 15 10 7)(3 6 11 14)(4 13 12 5)(17 20 25 28)(18 27 26 19)(21 32 29 24)(22 23 30 31)

G:=sub<Sym(32)| (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31)>;

G:=Group( (1,13,9,5)(2,14,10,6)(3,15,11,7)(4,16,12,8)(17,21,25,29)(18,22,26,30)(19,23,27,31)(20,24,28,32), (1,25)(2,26)(3,27)(4,28)(5,29)(6,30)(7,31)(8,32)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32), (1,8,9,16)(2,15,10,7)(3,6,11,14)(4,13,12,5)(17,20,25,28)(18,27,26,19)(21,32,29,24)(22,23,30,31) );

G=PermutationGroup([(1,13,9,5),(2,14,10,6),(3,15,11,7),(4,16,12,8),(17,21,25,29),(18,22,26,30),(19,23,27,31),(20,24,28,32)], [(1,25),(2,26),(3,27),(4,28),(5,29),(6,30),(7,31),(8,32),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32)], [(1,8,9,16),(2,15,10,7),(3,6,11,14),(4,13,12,5),(17,20,25,28),(18,27,26,19),(21,32,29,24),(22,23,30,31)])

Matrix representation of D4.5D8 in GL4(𝔽7) generated by

0651
3056
3361
1631
,
4320
6046
1145
0006
,
2045
5235
5214
4453
,
3633
6525
6102
2406
G:=sub<GL(4,GF(7))| [0,3,3,1,6,0,3,6,5,5,6,3,1,6,1,1],[4,6,1,0,3,0,1,0,2,4,4,0,0,6,5,6],[2,5,5,4,0,2,2,4,4,3,1,5,5,5,4,3],[3,6,6,2,6,5,1,4,3,2,0,0,3,5,2,6] >;

D4.5D8 in GAP, Magma, Sage, TeX

D_4._5D_8
% in TeX

G:=Group("D4.5D8");
// GroupNames label

G:=SmallGroup(128,955);
// by ID

G=gap.SmallGroup(128,955);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,736,422,1123,360,2804,718,172,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^8=d^2=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^7>;
// generators/relations

Export

Character table of D4.5D8 in TeX

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